Let R be a commutative ring. Let A 0 be an ideal in R[x] such that

the lowest degree of a nonzero polynomial in A is n 1 and such that A

contains a monic polynomial of degree n. Prove that A is a principal ideal

in R[x].

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- Apr 28th 2009, 01:46 PMmathman88Principle Ideal
Let R be a commutative ring. Let A 0 be an ideal in R[x] such that

the lowest degree of a nonzero polynomial in A is n 1 and such that A

contains a monic polynomial of degree n. Prove that A is a principal ideal

in R[x]. - Apr 28th 2009, 02:41 PMNonCommAlg
let be a monic polynomial of degree obviously now using induction over degrees of elements of we show that every elment of is divisible by which will

prove that so let then is a polynomial of degree at most and thus by the minimality of we have

thus this proves the base of the induction. now let and suppose every polynomial in of degree at most is in

let then and thus by induction hypothesis for some then